Cohen I.M., Kundu P.K. Fluid Mechanics

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Preface to Second Edition

My involvement with Pijush Kundu’s Fluid Mechanics ﬁrst began in April 1991 with

a letter from him asking me to consider his book for adoption in the ﬁrst year graduate

course I had been teaching for 25 years. That started a correspondence and, in fact,

I did adopt the book for the following academic year. The correspondence related

to improving the book by enhancing or clarifying various points. I would not have

taken the time to do that if I hadn’t thought this was the best book at the ﬁrst-year

graduate level. By the end of that year we were already discussing a second edition

and whether I would have a role in it. By early 1992, however, it was clear that I

had a crushing administrative burden at the University of Pennsylvania and could not

undertake any time-consuming projects for the next several years. My wife and I met

Pijush and Shikha for the ﬁrst time in December 1992. They were a charming, erudite,

sophisticated couple with two brilliant children. We immediately felt a bond of warmth

and friendship with them. Shikha was a teacher like my wife so the four of us had a

great deal in common. A couple of years later we were shocked to hear that Pijush had

died suddenly and unexpectedly. It saddened me greatly because I had been looking

forward to working with Pijush on the second edition after my term as department

chairman ended in mid-1997. For the next year and a half, however, serious family

health problems detoured any plans. Discussions on this edition resumed in July of

1999 and were concluded in the Spring of 2000 when my work really started. This

book remains the principal work product of Pijush K. Kundu, especially the lengthy

chapters on Gravity Waves, Instability, and Geophysical Fluid Dynamics, his areas of

expertise. I have added new material to all of the other chapters, often providing an

alternative point of view. Speciﬁcally, vector ﬁeld derivatives have been generalized,

as have been streamfunctions. Additional material has been added to the chapters on

laminar ﬂows and boundary layers. The treatment of one-dimensional gasdynamics

has been extended. More problems have been added to most chapters. Professor

Howard H. Hu, a recognized expert in computational ﬂuid dynamics, graciously

provided an entirely new chapter, Chapter 11, thereby providing the student with an

entree into this exploding new ﬁeld. Both ﬁnite difference and ﬁnite element methods

are introduced and a detailed worked-out example of each is provided.

I have been a student of ﬂuid mechanics since 1954 when I entered college to

study aeronautical engineering. I have been teaching ﬂuid mechanics since 1963 when

I joined the Brown University faculty, and I have been teaching a course corresponding

to this book since moving to the University of Pennsylvania in 1966. I am most grateful

to two of my own teachers, Professor Wallace D. Hayes (1918–2001), who expressed

ﬂuid mechanics in the clearest way I have ever seen, and Professor Martin D. Kruskal,

whose use of mathematics to solve difﬁcult physical problems was developed to a

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xxii Preface to Second Edition

high art form and reminds me of a Vivaldi trumpet concerto. His codiﬁcation of rules

of applied limit processes into the principles of “Asymptotology” remains with me

today as a way to view problems. I am grateful also to countless students who asked

questions, forcing me to rethink many points.

The editors at Academic Press, Gregory Franklin and Marsha Filion (assistant)

have been very supportive of my efforts and have tried to light a ﬁre under me. Since

this edition was completed, I found that there is even more new and original material

I would like to add. But, alas, that will have to wait for the next edition. The new ﬁgures

and modiﬁcations of old ﬁgures were done by Maryeileen Banford with occasional

assistance from the school’s software expert, Paul W. Shaffer. I greatly appreciate

their job well done.

Ira M. Cohen

Preface to First Edition

This book is a basic introduction to the subject of ﬂuid mechanics and is intended for

undergraduate and beginning graduate students of science and engineering. There is

enough material in the book for at least two courses. No previous knowledge of the

subject is assumed, and much of the text is suitable in a ﬁrst course on the subject. On

the other hand, a selection of the advanced topics could be used in a second course.

I have not tried to indicate which sections should be considered advanced; the choice

often depends on the teacher, the university, and the ﬁeld of study. Particular effort

has been made to make the presentation clear and accurate and at the same time easy

enough for students. Mathematically rigorous approaches have been avoided in favor

of the physically revealing ones.

A survey of the available texts revealed the need for a book with a balanced

view, dealing with currently relevant topics, and at the same time easy enough for

students. The available texts can perhaps be divided into three broad groups. One

type, written primarily for applied mathematicians, deals mostly with classical topics

such as irrotational and laminar ﬂows, in which analytical solutions are possible.

A second group of books emphasizes engineering applications, concentrating on

ﬂows in such systems as ducts, open channels, and airfoils. A third type of text is

narrowly focused toward applications to large-scale geophysical systems, omitting

small-scale processes which are equally applicable to geophysical systems as well as

laboratory-scale phenomena. Several of these geophysical ﬂuid dynamics texts are

also written primarily for researchers and are therefore rather difﬁcult for students.

I have tried to adopt a balanced view and to deal in a simple way with the basic ideas

relevant to both engineering and geophysical ﬂuid dynamics.

However, I have taken a rather cautious attitude toward mixing engineering and

geophysical ﬂuid dynamics, generally separating them in different chapters. Although

the basic principles are the same, the large-scale geophysical ﬂows are so dominated

by the effects of the Coriolis force that their characteristics can be quite different

from those of laboratory-scale ﬂows. It is for this reason that most effects of planetary

rotation are discussed in a separate chapter, although the concept of the Coriolis force

is introduced earlier in the book. The effects of density stratiﬁcation, on the other hand,

are discussed in several chapters, since they can be important in both geophysical and

laboratory-scale ﬂows.

The choice of material is always a personal one. In my effort to select topics,

however, I have been careful not to be guided strongly by my own research interests.

The material selected is what I believe to be of the most interest in a book on general

ﬂuid mechanics. It includes topics of special interest to geophysicists (for example,

the chapters on Gravity Waves and Geophysical Fluid Dynamics) and to engineers

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(for example, the chapters on Aerodynamics and Compressible Flow). There are also

chapters of common interest, such as the ﬁrst ﬁve chapters, and those on Boundary

Layers, Instability, and Turbulence. Some of the material is now available only in

specialized monographs; such material is presented here in simple form, perhaps

sacriﬁcing some formal mathematical rigor.

Throughout the book the convenience of tensor algebra has been exploited freely.

My experience is that many students feel uncomfortable with tensor notation in the

beginning, especially with the permutation symbol ε

ijk

. After a while, however, they

like it. In any case, following an introductory chapter, the second chapter of the book

explains the fundamentals of Cartesian Tensors. The next three chapters deal with

standard and introductory material on Kinematics, Conservation Laws, and Vorticity

Dynamics. Most of the material here is suitable for presentation to geophysicists as

well as engineers.

In much of the rest of the book the teacher is expected to select topics that are

suitable for his or her particular audience. Chapter 6 discusses Irrotational Flow; this

material is rather classical but is still useful for two reasons. First, some of the results

are used in later chapters, especially the one on Aerodynamics. Second, most of the

ideas are applicable in the study of other potential ﬁelds, such as heat conduction

and electrostatics. Chapter 7 discusses Gravity Waves in homogeneous and stratiﬁed

ﬂuids; the emphasis is on linear analysis, although brief discussions of nonlinear

effects such as hydraulic jump, Stokes’s drift, and soliton are given.

After a discussion of Dynamic Similarity in Chapter 8, the study of viscous ﬂow

starts with Chapter 9, which discusses Laminar Flow. The material is standard, but

the concept and analysis of similarity solutions are explained in detail. In Chapter 10

on Boundary Layers, the central idea has been introduced intuitively at ﬁrst. Only

after a thorough physical discussion has the boundary layer been explained as a sin-

gular perturbation problem. I ask the indulgence of my colleagues for including the

peripheral section on the dynamics of sports balls but promise that most students

will listen with interest and ask a lot of questions. Instability of ﬂows is discussed at

some length in Chapter 12. The emphasis is on linear analysis, but some discussion

of “chaos” is given in order to point out how deterministic nonlinear systems can lead

to irregular solutions. Fully developed three-dimensional Turbulence is discussed in

Chapter 13. In addition to standard engineering topics such as wall-bounded shear

ﬂows, the theory of turbulent dispersion of particles is discussed because of its geo-

physical importance. Some effects of stratiﬁcation are also discussed here, but the

short section discussing the elementary ideas of two-dimensional geostrophic turbu-

lence is deferred to Chapter 14. I believe that much of the material in Chapters 8–13

will be of general interest, but some selection of topics is necessary here for teaching

specialized groups of students.

The remaining three chapters deal with more specialized applications in geo-

physics and engineering. Chapter 14 on Geophysical Fluid Dynamics emphasizes

the linear analysis of certain geophysically important wave systems. However, ele-

ments of barotropic and baroclinic instabilities and geostrophic turbulence are also

included. Chapter 15 on Aerodynamics emphasizes the application of potential the-

ory to ﬂow around lift-generating proﬁles; an elementary discussion of ﬁnite-wing

theory is also given. The material is standard, and I do not claim much originality or

Preface to First Edition xxv

innovation, although I think the reader may be especially interested in the discussions

of propulsive mechanisms of ﬁsh, birds, and sailboats and the material on the historic

controversy between Prandtl and Lanchester. Chapter 16 on Compressible Flow also

contains standard topics, available in most engineering texts. This chapter is included

with the belief that all ﬂuid dynamicists should have some familiarity with such topics

as shock waves and expansion fans. Besides, very similar phenomena also occur in

other nondispersive systems such as gravity waves in shallow water.

The appendices contain conversion factors, properties of water and air, equations

in curvilinear coordinates, and short bibliographical sketches of Founders of Modern

Fluid Dynamics. In selecting the names in the list of founders, my aim was to come

up with a very short list of historic ﬁgures who made truly fundamental contributions.

It became clear that the choice of Prandtl and G. I. Taylor was the only one that would

avoid all controversy.

Some problems in the basic chapters are worked out in the text, in order to

illustrate the application of the basic principles. In a ﬁrst course, undergraduate engi-

neering students may need more practice and help than offered in the book; in that

case the teacher may have to select additional problems from other books. Difﬁcult

problems have been deliberately omitted from the end-of-chapter exercises. It is my

experience that the more difﬁcult exercises need a lot of clariﬁcation and hints (the

degree of which depends on the students’ background), and they are therefore better

Acknowledgments

I would like to record here my gratitude to those who made the writing of this book

possible. My teachers Professor Shankar Lal and Professor John Lumley fostered my

interest in ﬂuid mechanics and quietly inspired me with their brilliance; Professor

Lumley also reviewed Chapter 13. My colleague Julian McCreary provided support,

encouragement, and careful comments on Chapters 7, 12, and 14. Richard Thomson’s

cheerful voice over the telephone was a constant reassurance that professional science

can make some people happy, not simply competitive; I am also grateful to him for

reviewing Chapters 4 and 15. Joseph Pedlosky gave very valuable comments on

Chapter 14, in addition to warning me against too broad a presentation. John Allen

allowed me to use his lecture notes on perturbation techniques. Yasushi Fukamachi,

Hyong Lee, and Kevin Kohler commented on several chapters and constantly pointed

out things that may not have been clear to the students. Stan Middleman and Elizabeth

Mickaily were especially diligent in checking my solutions to the examples and

end-of-chapter problems. Terry Thompson constantly got me out of trouble with my

personal computer. Kathy Maxson drafted the ﬁgures. Chuck Arthur and Bill LaDue,

my editors at Academic Press, created a delightful atmosphere during the course of

writing and production of the book.

Lastly, I am grateful to Amjad Khan, the late Amir Khan, and the late Omkarnath

Thakur for their music, which made working after midnight no chore at all. I recom-

mend listening to them if anybody wants to write a book!

Pijush K. Kundu

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Author’s Notes

Both indicial and boldface notations are used to indicate vectors and tensors. The

comma notation to represent spatial derivatives (for example, A

for ∂A/∂x

) is used

in only two sections of the book (Sections 5.6 and 13.7), when the algebra became

cumbersome otherwise. Equal to by deﬁnition is denoted by ≡; for example, the

ratio of speciﬁc heats is introduced as γ ≡ C

. Nearly equal to is written as ,

proportional to is written as ∝, and of the order is written as ∼.

Plane polar coordinates are denoted by (r, θ), cylindrical polar coordinates are

denoted by either (R,ϕ,x)or (r, θ , x), and spherical polar coordinates are denoted by

(r, θ , ϕ) (see Figure 3.1). The velocity components in the three Cartesian directions

(x,y,z)are indicated by (u, v, w). In geophysical situations the z-axis points upward.

In some cases equations are referred to by a descriptive name rather than a number

(for example, “the x-momentum equation shows that ...”). Those equations and/or

results deemed especially important have been indicated by a box.

A list of literature cited and supplemental reading is provided at the end of most

chapters. The list has been deliberately kept short and includes only those sources that

serve one of the following three purposes: (1) it is a reference the student is likely to

ﬁnd useful, at a level not too different from that of this book; (2) it is a reference that

has inﬂuenced the author’s writing or from which a ﬁgure is reproduced; and (3) it

is an important work done after 1950. In currently active ﬁelds, reference has been

made to more recent review papers where the student can ﬁnd additional references

to the important work in the ﬁeld.

Fluid mechanics forces us fully to understand the underlying physics. This is

because the results we obtain often defy our intuition. The following examples support

these contentions:

1. Inﬁnitesmally small causes can have large effects (d’Alembert’s paradox).

2. Symmetric problems may have nonsymmetric solutions (von Karman vortex

street).

3. Friction can make the ﬂow go faster and cool the ﬂow (subsonic adiabatic ﬂow

in a constant area duct).

4. Roughening the surface of a body can decrease its drag (transition from laminar

to turbulent boundary layer separation).

5. Adding heat to a ﬂow may lower its temperature. Removing heat from a ﬂow

may raise its temperature (1-dimensional adiabatic ﬂow in a range of subsonic

Mach number).

6. Friction can destabilize a previously stable ﬂow (Orr-Sommerfeld stability

analysis for a boundary layer proﬁle without inﬂection point).

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7. Without friction, birds could not ﬂy and ﬁsh could not swim (Kutta condition

requires viscosity).

8. The best and most accurate visualization of streamlines in an inviscid (inﬁnite

Reynolds number) ﬂow is in a Hele-Shaw apparatus for creeping highly viscous

ﬂow (near zero Reynolds number).

Every one of these counterintuitive effects will be treated and discussed in

this text.

This second edition also contains additional material on streamfunctions, bound-

ary conditions, viscous ﬂows, boundary layers, jets, and compressible ﬂows. Most

important, there is an entirely new chapter on computational ﬂuid dynamics that intro-

duces the student to the various techniques for numerically integrating the equations

governing ﬂuid motions. Hopefully the introduction is sufﬁcient that the reader can

follow up with specialized texts for a more comprehensive understanding.

An historical survey of ﬂuid mechanics from the time of Archimedes (ca.

250 B.C.E.) to approximately 1900 is provided in the Eleventh Edition of

The Encyclopædia Britannica (1910) in Vol. XIV (under “Hydromechanics,”

pp. 115–135). I am grateful to Professor Herman Gluck (Professor of Mathemat-

ics at the University of Pennsylvania) for sending me this article. Hydrostatics and

classical (constant density) potential ﬂows are reviewed in considerable depth. Great

detail is given in the solution of problems that are now considered obscure and arcane

with credit to authors long forgotten. The theory of slow viscous motion developed by

Stokes and others is not mentioned. The concept of the boundary layer for high-speed

motion of a viscous ﬂuid was apparently too recent for its importance to have been

realized.

IMC

Chapter 1

Introduction

1. Fluid Mechanics ................... 1

2. Units of Measurement .............. 2

3. Solids, Liquids, and Gases ......... 3

4. Continuum Hypothesis ............. 4

5. Transport Phenomena ............. 5

6. Surface Tension.................... 8

7. Fluid Statics ....................... 9

Example 1.1...................... 12

8. Classical Thermodynamics ........ 12

First Law of Thermodynamics .... 13

Equations of State ................ 13

Speciﬁc Heats .................... 14

Second Law of Thermodynamics . 15

TdS Relations ................... 15

Speed of Sound.................. 16

Thermal Expansion Coefﬁcient . . 16

9. Perfect Gas ...................... 16

10. Static Equilibrium of

a Compressible Medium .......... 18

Potential Temperature and

Density ....................... 20

Scale Height of the Atmosphere . . 22

Exercises ........................ 22

Literature Cited ................. 24

Supplemental Reading ........... 24

1. Fluid Mechanics

Fluid mechanics deals with the ﬂow of ﬂuids. Its study is important to physicists,

whose main interest is in understanding phenomena. They may, for example, be

interested in learning what causes the various types of wave phenomena in the atmo-

sphere and in the ocean, why a layer of ﬂuid heated from below breaks up into cellular

patterns, why a tennis ball hit with “top spin” dips rather sharply, how ﬁsh swim, and

how birds ﬂy. The study of ﬂuid mechanics is just as important to engineers, whose

main interest is in the applications of ﬂuid mechanics to solve industrial problems.

Aerospace engineers may be interested in designing airplanes that have low resis-

tance and, at the same time, high “lift” force to support the weight of the plane. Civil

engineers may be interested in designing irrigation canals, dams, and water supply

systems. Pollution control engineers may be interested in saving our planet from the

constant dumping of industrial sewage into the atmosphere and the ocean. Mechan-

ical engineers may be interested in designing turbines, heat exchangers, and ﬂuid

couplings. Chemical engineers may be interested in designing efﬁcient devices to

mix industrial chemicals. The objectives of physicists and engineers, however, are

DOI: 10.1016/B978-0-12-381399-2.50001-0